proof of Hadamard three-circle theorem

For this, let α be a real number; the functionα⁢log⁡|z|+log⁡|f⁢(z)| is harmonic outside the zeros of f. Near the zeros of f the above function has values which are large negative. Hence by the maximum modulus principle this function has its maximum on the boundary of the annulus, specifically on the two circles |z|=r1 and |z|=r2. Therefore